At some point in your life you've almost certainly been taught $i = \sqrt{-1}$, which is fine if taken as a black box definition, but not particularly helpful in building intuition about operations on complex numbers. This slide shows a complex number represented as a vector in the complex (2D) plane. Notice how multiplication by $i$ corresponds to rotation by 90 degrees in the plane, and multiplication by $i^2$ results in a rotation by 180 degrees, or equivalently, scaling of the original vector by -1.

BryceSummers

The property wherebye multiplication by complex numbers of magnitude 1 results in rotations is clearly demonstrated by Euler's formula:

$e^{ix}=\cos x+i\sin x$

We can convert any two normalized complex numbers $\cos(x_{1}) + i\sin(x_{1})$ and $\cos(x_{2}) + i\sin(x_{2})$ to numbers to the form $e^{ix_{1}}$ and $e^{ix_{2}}$, then the multiplication results in the value $e^{i(x_{1} + x_{2})}$ which equals $\cos(x_{1} + x_{2}) + i \sin(x_{1} + x_{2})$

Thus every complex number multiplication results in a change in magnitude and a linear change in radial coordinate, and the change of magnitude can be ignored for complex numbers of magnitude 1.

At some point in your life you've almost certainly been taught $i = \sqrt{-1}$, which is fine if taken as a black box definition, but not particularly helpful in building intuition about operations on complex numbers. This slide shows a complex number represented as a vector in the complex (2D) plane. Notice how multiplication by $i$ corresponds to rotation by 90 degrees in the plane, and multiplication by $i^2$ results in a rotation by 180 degrees, or equivalently, scaling of the original vector by -1.

The property wherebye multiplication by complex numbers of magnitude 1 results in rotations is clearly demonstrated by Euler's formula:

$e^{ix}=\cos x+i\sin x$

We can convert any two normalized complex numbers $\cos(x_{1}) + i\sin(x_{1})$ and $\cos(x_{2}) + i\sin(x_{2})$ to numbers to the form $e^{ix_{1}}$ and $e^{ix_{2}}$, then the multiplication results in the value $e^{i(x_{1} + x_{2})}$ which equals $\cos(x_{1} + x_{2}) + i \sin(x_{1} + x_{2})$

Thus every complex number multiplication results in a change in magnitude and a linear change in radial coordinate, and the change of magnitude can be ignored for complex numbers of magnitude 1.